文章目录
- [1. 数学模型](#1. 数学模型)
- [2. Wirtinger Flow 算法](#2. Wirtinger Flow 算法)
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- [2.1. 光谱初始化方法](#2.1. 光谱初始化方法)
- [2.2. Wirtinger梯度下降](#2.2. Wirtinger梯度下降)
- [3. 算法实现](#3. 算法实现)
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- [3.1. Matlab实现](#3.1. Matlab实现)
- [3.2. Python实现](#3.2. Python实现)
- 参考文献
1. 数学模型
观测数学模型可由下面公式给出
y = ∣ A x ∣ 2 y = |Ax|^2 y=∣Ax∣2
其中 x ∈ C n x\in\mathbb C^{n} x∈Cn， A ∈ C m × n A\in\mathbb C^{m\times n} A∈Cm×n， y ∈ R m y\in\mathbb R^{m} y∈Rm。
所以我们要求解的问题可归为如下非凸最小二乘问题
min z ∈ C n f ( z ) = 1 2 ∥ ∣ A z ∣ 2 − y ∥ 2 2 \min_{z\in\mathbb C^{n}} \;f(z)=\frac12\Bigl\|\,|Az|^{ 2}-y\Bigr\|_2^{2} z∈Cnminf(z)=21 ∣Az∣2−y 22
2. Wirtinger Flow 算法
该算法可以总结成两步：1. 光谱初始化 2. Wirtinger梯度下降
2.1. 光谱初始化方法
具体步骤如下：
-
能量标定系数
λ 2 = n 1 m ⊤ y ∥ A ∥ F 2 \lambda^{2} \;=\; n\,\frac{\mathbf 1_{m}^{\!\top}y}{\|A\|_{F}^{2}} λ2=n∥A∥F21m⊤y -
构造自伴矩阵
Y = 1 m A ∗ diag ( y ) A Y \;=\; \frac1m\,A^{*}\operatorname{diag}(y)\,A Y=m1A∗diag(y)A求得其最大特征向量 v v v
-
缩放得到初始点
z 0 = λ v z_{0}=\lambda\,v z0=λv
2.2. Wirtinger梯度下降
更新公式如下：
z τ + 1 = z τ − μ τ + 1 ∥ z 0 ∥ 2 2 ( 1 m A ∗ [ ( ∣ A z τ ∣ 2 − y ) ⊙ ( A z τ ) ] ) ⏟ ∇ f ( z τ ) z_{\tau+1} =\;z_\tau\;-\;\frac{\mu_{\tau+1}}{\|z_0\|{2}^{2}}\; \underbrace{\Bigl(\frac1m\,A^{*}\bigl[\,(|A z\tau|^{2}-y)\;⊙\;(A z_\tau)\bigr]\Bigr)}{\nabla f(z\tau)} zτ+1=zτ−∥z0∥22μτ+1∇f(zτ) (m1A∗[(∣Azτ∣2−y)⊙(Azτ)])
公式中的 μ \mu μ更新根据经验公式
μ τ = min ( 1 − exp ( − τ / τ 0 ) , 0.2 ) , τ 0 ≈ 330 \mu_\tau=\min(1-\exp(-\tau/\tau_0),\,0.2),\quad \tau_0≈330 μτ=min(1−exp(−τ/τ0),0.2),τ0≈330
3. 算法实现
3.1. Matlab实现
clear; close all; clc
%% Measurement model
% Signal length
n = 128;
% Complex signal
x = randn(n,1) + 1i*randn(n,1);
% measurement number
m = 5 * n;
% Measurement matrix
A = 1/sqrt(2)*randn(m,n) + 1i/sqrt(2)*randn(m,n);
% Measured values
y = abs(A*x).^2 ;
%% Initialization
% power method to get the initial guess
npower_iter = 50;
% Scaled coefficient lambda
lam = sqrt(n * sum(y) / norm(A, 'fro')^2);
% Random input
z0 = randn(n,1); z0 = z0/norm(z0,'fro');
for tt = 1:npower_iter
z0 = 1/m * A'*(y .* (A*z0));
z0 = z0/norm(z0,'fro');
end
% Initialized ouput
z = lam * z0;
%% Gradient update
% Max number of iterations
max_iter = 2500;
% update mu
tau0 = 330;
mu = @(t) min(1-exp(-t/tau0), 0.2);
% Store relative errors
relative_error = zeros(max_iter, 1);
for tt = 1:max_iter
Az = A*z;
% Wirtinger gradient
grad = 1/m* A'*( ( abs(Az).^2-y ) .* Az );
% ||z0||=lam
z = z - mu(tt)/lam^2 * grad;
% Calculate relative error value
relative_error_val = norm(x - exp(-1i*angle(trace(x'*z))) * z, 'fro')/norm(x,'fro');
relative_error(tt) = relative_error_val;
end
%%
figure,semilogy(relative_error,'LineWidth',1.8, 'Color',[0 0.4470 0.7410])
xlabel('Iteration','FontSize',16,'FontName','Times New Roman')
ylabel('Relative error','FontSize',16,'FontName','Times New Roman')
title('Wirtinger Flow Convergence','FontSize',16,'FontWeight','bold')
3.2. Python实现
import numpy as np
import matplotlib.pyplot as plt
n = 128 # 信号长度
x = np.random.randn(n) + 1j * np.random.randn(n) # 复值真信号
m = 5 * n # 测量数
A = (np.random.randn(m, n) + 1j * np.random.randn(m, n)) / np.sqrt(2)
y = np.abs(A @ x) ** 2 # 强度观测 |Ax|^2
# ---------- Initialization (power method) ------------------------------------
npower_iter = 50 # 幂迭代次数
lam = np.sqrt(n * y.sum() / np.linalg.norm(A, "fro") ** 2) # λ
z0 = np.random.randn(n) + 1j * np.random.randn(n)
z0 /= np.linalg.norm(z0)
for _ in range(npower_iter):
z0 = (A.conj().T @ (y * (A @ z0))) / m
z0 /= np.linalg.norm(z0)
z = lam * z0 # 初值
# ---------- Gradient update ---------------------------------------------------
max_iter = 2500
tau0 = 330.0
rel_err = np.zeros(max_iter)
for tt in range(max_iter):
mu = min(1.0 - np.exp(-(tt + 1) / tau0), 0.2) # 步长 μ_t
Az = A @ z
grad = (A.conj().T @ ((np.abs(Az) ** 2 - y) * Az)) / m
z = z - (mu / lam ** 2) * grad
# 相对误差
theta = np.angle(np.vdot(x, z)) # vdot = x* · z
rel_err[tt] = np.linalg.norm(x - np.exp(-1j * theta) * z) / np.linalg.norm(x)
# ---------- plot -------------------------------------------------------
plt.figure(figsize=(6.2, 4.2), facecolor="w")
plt.semilogy(rel_err, lw=1.8, color=(0.0, 0.447, 0.741))
plt.xlabel("Iteration", fontsize=13)
plt.ylabel("Relative error", fontsize=13)
plt.title("Wirtinger Flow Convergence (1-D)", fontsize=15, weight="bold")
plt.grid(ls="--", alpha=0.3)
plt.tight_layout()
plt.show()
参考文献
Candes E J, Li X, Soltanolkotabi M. Phase retrieval via Wirtinger flow: Theory and algorithms[J]. IEEE Transactions on Information Theory, 2015, 61(4): 1985-2007.